Poisson Processes in Free Probability

TL;DR

This paper develops a multidimensional free Poisson limit theorem, defines joint free Poisson distributions, constructs free Poisson processes, and extends the theory to free Poisson random measures with an associated integration framework.

Contribution

It introduces a comprehensive framework for free Poisson processes and measures, including their construction, properties, and integration theory, advancing free probability theory.

Findings

Proved a multidimensional free Poisson limit theorem.

Defined joint free Poisson distributions in non-commutative spaces.

Developed an integration theory for free Poisson random measures.

Abstract

We prove a multidimensional Poisson limit theorem in free probability, and define joint free Poisson distributions in a non-commutative probability space. We define (compound) free Poisson process explicitly, similar to the definitions of (compound) Poisson processes in classical probability. We proved that the sum of finitely many freely independent compound free Poisson processes is a compound free Poisson processes. We give a step by step procedure for constructing a (compound) free Poisson process. A Karhunen-Loeve expansion theorem for centered free Poisson processes is proved. We generalize free Poisson processes to a notion of free Poisson random measures (which is slightly different from the previously defined ones in free probability, but more like an analogue of classical Poisson random measures). Then we develop the integration theory of real-valued functions with respect to a free Poisson random measure, generalizing the classical integration theory to the free probability case. We find that the integral of a function (in certain spaces of functions) with respect to a free Poisson random measure has a compound free Poisson distribution. For centered free Poisson random measures, we can get a simpler and more beautiful integration theory.